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%I #16 May 04 2019 21:49:14
%S 1,0,2,1,2,1,2,2,3,2,2,2,2,2,4,3,2,3,2,2,4,2,2,4,3,2,4,4,2,3,2,4,4,2,
%T 4,5,2,2,4,4,2,3,2,4,6,2,2,6,3,4,4,4,2,5,4,4,4,2,2,6,2,2,6,5,4,5,2,4,
%U 4,6,2,6,2,2,6,4,4,5,2,6,5,2,2,6,4,2,4,6,2,5,4,4,4,2,4,8,2,4,6,5,2,5,2,6,8
%N a(n) is the number of isolated divisors of n.
%C A divisor d of n is isolated if neither d-1 nor d+1 divides n.
%C The convention for 1 is that it is an isolated divisor iff n is odd. - _Olivier Gérard_, Sep 22 2007
%H Ray Chandler, <a href="/A132881/b132881.txt">Table of n, a(n) for n=1..10000</a>
%F a(n) = A000005(n) - A132747(n).
%e The positive divisors of 56 are 1,2,4,7,8,14,28,56. Of these, 1 and 2 are adjacent and 7 and 8 are adjacent. The isolated divisors are therefore 4,14,28,56. There are 4 of these, so a(56) = 4.
%p with(numtheory): a:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1, div)=false and member(div[i]+1, div)=false then ISO:=`union`(ISO,{div[i]}) end if end do end proc; 1, 0, seq(nops(a(j)), j=3..105); # _Emeric Deutsch_, Oct 02 2007
%t Table[Length@Select[Divisors[n],(#==1||Mod[n,#-1]>0)&&Mod[n,#+1]>0&],{n,1,200}] - _Olivier Gérard_ Sep 22 2007.
%t id[n_]:=DivisorSigma[0,n]-Length[Union[Flatten[Select[Partition[Divisors[ n],2,1],#[[2]]-#[[1]]==1&]]]]; Array[id,110] (* _Harvey P. Dale_, Jun 04 2018 *)
%Y Cf. A132882, A132747.
%K nonn
%O 1,3
%A _Leroy Quet_, Sep 03 2007
%E More terms from _Olivier Gérard_, Sep 22 2007
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